The processing and interpretation of multicomponent (3C/4C) seismic data, acquired directly at the seafloor or during 3C land seismic surveys is compromised by the effects that the shallow subsurface has on the deeper reflected wavefield. The near surface is generally associated with low, laterally varying shear wave velocities and on land, the P-wave velocity can also be low. These properties often lead to large P- and S-wave traveltime perturbations in the deeper reflected wavefield which vary from receiver to receiver. In addition, there are suggestions that scattering and an elastic attenuation (especially shear) impacts amplitudes and recorded waveforms as well.
The receiver function methodology has it roots in earthquake seismology, where it was developed to investigate the structure of the crust and upper mantle using multicomponent teleseismic body wave data. Certain aspects of receiver functions are described for example by T. Ryberg and M. Weber in Geophys. J. Int. 141 (2000), 1-11. Reference to aspects of receiver function can also be found in the published patent applications GB 2384557 and WO 02/059647.
While trying to image discontinuities in upper mantle using receiver functions, it was realised that the change in the traveltime difference between the P-wave and the PS-converted wave, as a function of the angle of incidence, could no longer be neglected. T. Ryberg and M. Weber proposed to calculate a velocity spectrum stack (VSS) based on the change in the traveltime difference (hereafter called moveout) in a single layer to find optimal velocities for stacking of receiver functions. Although this approach gives good results when the structure is relatively homogeneous above the converting interface, it lacks a theoretical basis when applied to layered media with significant other discontinuities above the converting interface, i.e. when ray bending is likely to produce significant deviations from the traveltime difference equation for a single layer. This was partly recognised and, subsequently, it was suggested to substitute the average vertical P- and S-wave slowness for a stack of layers, into the equation for a single layer. It was also proposed to circumvent this problem by assuming a reference model (providing an initial vp(z) and vs(z)) and a corresponding one-parameter family of related models by multiplying the reference model by a fraction close to one, and obtain the difference in traveltime as a function of slowness by ray-tracing through the perturbed reference model. This approach severely restricts the number of free parameters.
On the other hand, normal moveout correction (NMO), velocity analysis and stacking of multichannel data has a long history in exploration and production seismic and the theory is well developed for horizontally layered media. It was demonstrated how to derive a power series approximation of the squared reflection traveltime as a function of even powers of offset (i.e. of the form: t2≈c1+c2·x2+c3·X4+ . . . , where t denotes two-way traveltime, x denotes offset and the coefficients depend on layer thicknesses and velocities of the medium). Hyperbolic approximations were used, truncating the infinite series after the second term and defined the rms-velocity as the square-root of the inverse of coefficient c2. It is also known that the slope of the x2−t2 curve at x=0 yields the inverse of the squared rms-velocity and how from the rms-velocity at two consecutive depth levels the interval velocity between them can be calculated. Taner and Koehler in: Geophysics 34 (1969), 859-881 also introduced the velocity spectrum stacking technique using a multichannel coherence measure called semblance. The work by Taner and Koehler was generalised for PS-converted waves by Tessmer and Behle in Geophys. Prosp. 36 (1988), 671-688, who also derive a Dix-Krey type formula, relating rms-velocities for PS-converted waves to products of P- and S-wave interval velocities in each layer.
In view of the above state of art, it is an object of the present invention to extend and improve the use of receiver functions to process and interpret seismic data to derive images of an earth. It is further object of the invention to determine velocities or velocity models from such receiver functions.